if a=b^x,b=c^y and c= a^z prove that xyz =1 ​

Question

if a=b^x,b=c^y and c= a^z prove that xyz =1

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2 months 2021-11-16T07:50:28+00:00 2 Answers 0 views 0

see explanation

Step-by-step explanation:

Using the rule of logarithms

log = nlogx

Given

a = ← take the log of both sides

loga = log = xlogb ⇒ x = b = ← take the log of both sides

logb = log = ylogc ⇒ y = c = ← take the log of both sides

logc = log = zloga ⇒ z = Thus

xyz = × × ← cancel loga, logb, logc on numerators/denominators

Hence xyz = 1

Step-by-step explanation:

a = b^x given

log a = log b^x

log a = x log b

divide both sides by log b

x = log a/ log b … (1)

b = c^y given

log b = log c^y

log b = y log c

divide both sides by log c

y = log b/ log c … (2)

c = a^z given

introducing log to both sides

log c = log a^z

log c = z log a

divide both sides by log a

z = log c / log a … (3)

Multiplying equations 1, 2 and 3 together;

x*y *z = (log a/ log b) * (log b/ log c)

* ( log c/ log a)

Therefore,

xyz = 1. proved